The problem with irony is that most people think that, as Alanis Morrissette claimed, irony is rain on a wedding day, an acrophobic person dying on their first airplane trip, or an old man that dies the day after hitting the lottery, when in fact none of them are "ironic".
They are unfortunate happenstances.
What is actually truly Ironic about the song in question is the fact that she wrote it thinking it would be chock full of good examples of irony, when it instead contains none at all.

Grammar nazis had better stay away from here.
And on that note, in the interest of keeping the peace, I will not be back to this thread.

Are you one of those people that think irony and sarcasm are synonymous?

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o.O Hardly.

How did you get that conclusion?

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That was totally random, actually

It's just that, everyone has some weird definition of "irony" (including the one saying it's "synonymous with sarcasm", when used in speech, which is quite absurd), even dictionaries have conflicting definitions. Often within one single dictionary.

* S: (n) sarcasm, irony, satire, caustic remark (witty language used to convey insults or scorn) "he used sarcasm to upset his opponent"; "irony is wasted on the stupid"; "Satire is a sort of glass, wherein beholders do generally discover everybody's face but their own"--Jonathan Swift
* S: (n) irony (incongruity between what might be expected and what actually occurs) "the irony of Ireland's copying the nation she most hated"
* S: (n) irony (a trope that involves incongruity between what is expected and what occurs)

I mean, that could be anything, ever. Including, but not limited to, most of the examples from Alanis' song. I.e., unfortunate happenstances can be ironical, in context.

And the only definition that wouldn't cover at least half of the song would be the first one, saying that irony is actually sarcasm.